Problem: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{7z^3 - 14z^2 - 245z}{-8z^2 + 32z + 168}$
First factor out the greatest common factors in the numerator and in the denominator. $ q = \dfrac {7z(z^2 - 2z - 35)} {-8(z^2 - 4z - 21)} $ $ q = -\dfrac{7z}{8} \cdot \dfrac{z^2 - 2z - 35}{z^2 - 4z - 21} $ Next factor the numerator and denominator. $ q = - \dfrac{7z}{8} \cdot \dfrac{(z - 7)(z + 5)}{(z - 7)(z + 3)}$ Assuming $z \neq 7$ , we can cancel the $z - 7$ $ q = - \dfrac{7z}{8} \cdot \dfrac{z + 5}{z + 3}$ Therefore: $ q = \dfrac{ -7z(z + 5)}{ 8(z + 3)}$, $z \neq 7$